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Properties

Label 2-475-19.7-c1-0-19
Degree 22
Conductor 475475
Sign 0.989+0.146i0.989 + 0.146i
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.155 + 0.269i)2-s + (0.514 + 0.891i)3-s + (0.951 − 1.64i)4-s + (−0.160 + 0.277i)6-s + 3.28·7-s + 1.21·8-s + (0.969 − 1.67i)9-s − 5.16·11-s + 1.95·12-s + (1.76 − 3.06i)13-s + (0.510 + 0.883i)14-s + (−1.71 − 2.96i)16-s + (−0.504 − 0.874i)17-s + 0.603·18-s + (2.42 + 3.62i)19-s + ⋯
L(s)  = 1  + (0.109 + 0.190i)2-s + (0.297 + 0.514i)3-s + (0.475 − 0.824i)4-s + (−0.0653 + 0.113i)6-s + 1.23·7-s + 0.429·8-s + (0.323 − 0.559i)9-s − 1.55·11-s + 0.565·12-s + (0.490 − 0.849i)13-s + (0.136 + 0.236i)14-s + (−0.428 − 0.742i)16-s + (−0.122 − 0.211i)17-s + 0.142·18-s + (0.555 + 0.831i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.989+0.146i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.989+0.146i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.989+0.146i0.989 + 0.146i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(26,⋅)\chi_{475} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), 0.989+0.146i)(2,\ 475,\ (\ :1/2),\ 0.989 + 0.146i)

Particular Values

L(1)L(1) ≈\approx 2.00058−0.147218i2.00058 - 0.147218i
L(12)L(\frac12) ≈\approx 2.00058−0.147218i2.00058 - 0.147218i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1+(−2.42−3.62i)T 1 + (-2.42 - 3.62i)T
good2 1+(−0.155−0.269i)T+(−1+1.73i)T2 1 + (-0.155 - 0.269i)T + (-1 + 1.73i)T^{2}
3 1+(−0.514−0.891i)T+(−1.5+2.59i)T2 1 + (-0.514 - 0.891i)T + (-1.5 + 2.59i)T^{2}
7 1−3.28T+7T2 1 - 3.28T + 7T^{2}
11 1+5.16T+11T2 1 + 5.16T + 11T^{2}
13 1+(−1.76+3.06i)T+(−6.5−11.2i)T2 1 + (-1.76 + 3.06i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.504+0.874i)T+(−8.5+14.7i)T2 1 + (0.504 + 0.874i)T + (-8.5 + 14.7i)T^{2}
23 1+(3.83−6.63i)T+(−11.5−19.9i)T2 1 + (3.83 - 6.63i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.01+3.48i)T+(−14.5−25.1i)T2 1 + (-2.01 + 3.48i)T + (-14.5 - 25.1i)T^{2}
31 1+4.60T+31T2 1 + 4.60T + 31T^{2}
37 1−6.48T+37T2 1 - 6.48T + 37T^{2}
41 1+(−3.40−5.89i)T+(−20.5+35.5i)T2 1 + (-3.40 - 5.89i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.15−5.46i)T+(−21.5+37.2i)T2 1 + (-3.15 - 5.46i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.92−3.32i)T+(−23.5−40.7i)T2 1 + (1.92 - 3.32i)T + (-23.5 - 40.7i)T^{2}
53 1+(3.55−6.16i)T+(−26.5−45.8i)T2 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2}
59 1+(6.73+11.6i)T+(−29.5+51.0i)T2 1 + (6.73 + 11.6i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.06−5.31i)T+(−30.5−52.8i)T2 1 + (3.06 - 5.31i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.59−9.69i)T+(−33.5−58.0i)T2 1 + (5.59 - 9.69i)T + (-33.5 - 58.0i)T^{2}
71 1+(−0.227−0.394i)T+(−35.5+61.4i)T2 1 + (-0.227 - 0.394i)T + (-35.5 + 61.4i)T^{2}
73 1+(−2.06−3.57i)T+(−36.5+63.2i)T2 1 + (-2.06 - 3.57i)T + (-36.5 + 63.2i)T^{2}
79 1+(1.44+2.50i)T+(−39.5+68.4i)T2 1 + (1.44 + 2.50i)T + (-39.5 + 68.4i)T^{2}
83 1−5.50T+83T2 1 - 5.50T + 83T^{2}
89 1+(−3.56+6.17i)T+(−44.5−77.0i)T2 1 + (-3.56 + 6.17i)T + (-44.5 - 77.0i)T^{2}
97 1+(5.41+9.37i)T+(−48.5+84.0i)T2 1 + (5.41 + 9.37i)T + (-48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.90468054882854126873713389160, −10.12314142471837672138831849052, −9.444965805474070574207145383346, −7.940209812471432824888955558583, −7.67742795396989195145752606347, −6.04945101508224644405423215787, −5.35395011925648556211977415357, −4.38383642057555566271134348057, −2.89784213070802412340675698576, −1.39551652306263623325141105058, 1.86639212299655770787017072433, 2.65104240519539155850406982814, 4.25384563231141059121725535095, 5.13301105467250053010815389269, 6.67623642801789357102862226930, 7.65348455102235334405070065600, 8.035168817810422737299512588345, 8.923387058130292895547825483928, 10.68045350644073893581626534364, 10.86823690174494398365189872506

Graph of the ZZ-function along the critical line